3.165 \(\int k^{x^2} x \, dx\)

Optimal. Leaf size=13 \[ \frac{k^{x^2}}{2 \log (k)} \]

[Out]

k^x^2/(2*Log[k])

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Rubi [A]  time = 0.0072021, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2209} \[ \frac{k^{x^2}}{2 \log (k)} \]

Antiderivative was successfully verified.

[In]

Int[k^x^2*x,x]

[Out]

k^x^2/(2*Log[k])

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int k^{x^2} x \, dx &=\frac{k^{x^2}}{2 \log (k)}\\ \end{align*}

Mathematica [A]  time = 0.0016624, size = 13, normalized size = 1. \[ \frac{k^{x^2}}{2 \log (k)} \]

Antiderivative was successfully verified.

[In]

Integrate[k^x^2*x,x]

[Out]

k^x^2/(2*Log[k])

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Maple [A]  time = 0.003, size = 12, normalized size = 0.9 \begin{align*}{\frac{{k}^{{x}^{2}}}{2\,\ln \left ( k \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(k^(x^2)*x,x)

[Out]

1/2*k^(x^2)/ln(k)

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Maxima [A]  time = 0.941191, size = 15, normalized size = 1.15 \begin{align*} \frac{k^{\left (x^{2}\right )}}{2 \, \log \left (k\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(k^(x^2)*x,x, algorithm="maxima")

[Out]

1/2*k^(x^2)/log(k)

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Fricas [A]  time = 1.73341, size = 27, normalized size = 2.08 \begin{align*} \frac{k^{\left (x^{2}\right )}}{2 \, \log \left (k\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(k^(x^2)*x,x, algorithm="fricas")

[Out]

1/2*k^(x^2)/log(k)

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Sympy [A]  time = 0.095293, size = 17, normalized size = 1.31 \begin{align*} \begin{cases} \frac{k^{x^{2}}}{2 \log{\left (k \right )}} & \text{for}\: 2 \log{\left (k \right )} \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(k**(x**2)*x,x)

[Out]

Piecewise((k**(x**2)/(2*log(k)), Ne(2*log(k), 0)), (x**2/2, True))

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Giac [A]  time = 1.08108, size = 15, normalized size = 1.15 \begin{align*} \frac{k^{\left (x^{2}\right )}}{2 \, \log \left (k\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(k^(x^2)*x,x, algorithm="giac")

[Out]

1/2*k^(x^2)/log(k)